# Thread: Axiom of Union and Powerset

1. ## Axiom of Union and Powerset

Any thoughts on how to solve this one?

Prove that for all x, UP(x) = x

2. Hi,
it is only about verifying definitions:
$\displaystyle y \in x$ implies $\displaystyle \{y\} \in \mathcal{P}(x)$, and this implies $\displaystyle y \in \cup \mathcal{P}(x)$ (because union of a set contains elemets of the set's elements).
On the other hand,
$\displaystyle y \in \cup \mathcal{P}(x)$ implies $\displaystyle \left(\exists z\right) y \in z \& z \in \mathcal{P}(x)$, and this implies $\displaystyle y \in x$ (because $\displaystyle z \in \mathcal{P}(x)$ means $\displaystyle z \subseteq x$).